Universal quantum computation via scalable measurement-free error correction

TL;DR

The work advances universal fault-tolerant quantum computation without mid-circuit measurements by introducing two complementary measurement-free strategies: a deformation-based protocol on the Bacon-Shor code to realize a logical $CCZ$ gate and a concatenation scheme aided by a disposable Toffoli gadget. It demonstrates a circuit-level depolarizing threshold of about $1.2\times 10^{-3}$ for the concatenated Bacon-Shor architecture and provides efficient simulation tools for non-Clifford operations under noise. The methods address both gate- and error-correction processes in a measurement-free setting, offering a scalable path to fault-tolerance with potential practical impact for neutral-atom platforms. Together, these results lay out a blueprint for FT quantum computation that minimizes feed-forward and measurement latency while maintaining strong error suppression.

Abstract

We show that universal quantum computation can be concretely made fault-tolerant without mid-circuit measurements. To this end, we introduce a measurement-free deformation protocol of the Bacon-Shor code to realize a logical $\mathit{CCZ}$ gate. Combined with a fold-transversal logical Hadamard gate, this enables a universal set of fault-tolerant operations using only transversal gates and qubit permutations. For the purpose of benchmarking under circuit-level noise, we develop an efficient method to simulate non-Clifford circuits with a small number of Hadamard gates. Separately, we demonstrate that certain CSS codes can be concatenated without measurements or having to rely on a universal logical gate set. This is made possible by means of a resource-efficient gadget -- termed the ``disposable Toffoli gadget'' -- that realizes the error-correcting feedback. Then, under concatenation of the Bacon-Shor code, we observe a fault-tolerance threshold at a circuit-level depolarizing noise rate of approximately $0.12\,\%$. Together, the deformation and concatenation protocols outline a blueprint for a fully fault-tolerant architecture without any feed-forward operation, particularly suited to state-of-the-art neutral-atom platforms.

Figures (11)

• Figure 1: (a) The $3 \times 3$ Bacon Shor code, with red (blue) areas depicting the support of $X$-type ($Z$-type) stabilizer operators. (b) Any Bacon-Shor code has a gauge freedom. In particular, the pairwise $Z_i Z_j$ gauges along rows can be chosen to be $+1$, resulting in a Shor code. We call this the Shor gauge. To perform a logical $\mathit{CCZ}$, each logical qubit must start from this gauge, and then be extended to a $3 \times 9$ Bacon-Shor code, as shown in (c). The logical $\mathit{CCZ}$ is then performed between logical qubits, by a transversal application of physical $\mathit{CCZ}$ gates, with permutations across the columns. (d) To return to the original $3 \times 3$ configuration, a shrink move is performed. (e) To increase the tolerance to errors, concatenation of the same code is performed, where the data qubits of a code are composed of logical qubits at a lower level.
• Figure 2: (a) The logical operators of a Bacon-Shor code correspond to a single row ($X_L$) or column ($Z_L$). (b) A bit-flip repetition code is formed by pairwise $S^R_i = Z_{i} Z_{i + 1}$ stabilizers along a chain. (c) A transversal $\mathit{CCX}$ between a three-qubit repetition code and $d=3$ Bacon-Shor code. Note that this gate is only unidirectional. (d)--(e) The unencoding operation $\Lambda_Z$ ($\Lambda_X$) maps a Bacon-Shor code down to a bit-flip (phase-flip) repetition code. Intuitively, each column (row), supporting a separate representation of the relevant logical operator, is mapped to a qubit of the resulting repetition code. The stabilizers after each layer of gates are depicted.
• Figure 3: Graphical depiction of the connectivity required for the transversal $\mathit{CCZ}_L \ket{\psi_1 \psi_2 \psi_3}_L$ between three logical states $\ket{\psi_k}_L$, $k=1,2,3$, each encoded in the $3\times 9$ the Bacon-Shor codes, in the Shor gauge. Different colors correspond to different $\mathit{CCZ}$ gates between rows, and highlight that the permutations occur only across columns.
• Figure 4: (a) Components of the protocol for the $\mathit{CCZ}_L$ gate between $3 \times 3$ Bacon-Shor codes. (b) Measurent-free gauge fixing by teleportation. A logical Bell state $(\ket{00}_L + \ket{11}_L) / \sqrt{2}$ is first prepared in the Shor gauge. Note that, because the $\ket{+}_L$ is prepared in the Shor gauge, the last logical qubit does not need to be prepared in $\ket{0}_L$. This auxiliary state is then entangled with the logical state, and then MF teleportation using $\mathit{CX}$ and $\mathit{CZ}$ gates is performed, by first unencoding to the repetition code, cf. \ref{['sec:bacon-shor-repetition']}. (c) The extend gadget deforms a $3 \times 3$ Bacon-Shor code to its $3 \times 9$ variant, by exploiting the Shor gauge in the previous step. (d) A measurement-free error-correction round for the repetition code, used to correct for single bit flips on each triplet of each row of the logical states. (e) The extend gadget is reversed by a shrink gadget, bringing the logical state back to a $3 \times 3$ Bacon-Shor code.
• Figure 5: Average error rate of the logical $\mathit{CCZ}$ gate under depolarizing noise, computed using \ref{['eq:fidelity-ccz']}. The performance is benchmarked for initial states encoded without errors in the Shor (blue) or anti-Shor (red) gauges. Starting from $\ket{+++}_L$, we can prepare the non-Clifford resource state $\ket{CCZ}_L$. We therefore show the fidelity (orange) of its preparation using a noisy FT preparation in the Shor gauge, and without any gauge-fixing. Shaded regions correspond to $\qty{99}{\percent}$ confidence intervals.